How can you launch an object at a specific velocity and angle outside of the game engine?
Faceb Faceb has most of the solution.  The only bit he left out is that you also keyframe the Animated checkbox.  You have it checked at the beginning, but just before the motion of the Location keyframes are completed, you add a keyframe that turns off the Animated checkbox.

That deactivates the animation system and lets the physics system take over and it uses the velocity left over from the animation system



I have keyframed Suzanne's X and Z location.  You might also keyframe Y.  Notice also the channel called Kinematic (Rigid Body Object) which is the fcurve for the Animated checkbox that appears green in the Physics > Rigid Body section (green or yellow means keyframed)

To know the velocity you need to know the derivative of the location.  Since I altered my keyframes to have linear interpolation it is a simple matter of vx=dx/dt which in this case becomes vx=(1-0)/( (6-1) / 24fps ) = 4.8 and vz=(3-2)/( (6-1) / 24 fps ) = 4.8 and the total velocity would be sqrt(4.8**2 + 4.8**2) = 6.7882 .  The angle would be atan((1-0)/(3-2)) = 45 degrees .  

If you want to choose your own velocity and angle it is probably easiest to parent Suzanne to an empty.  Rig the empty with a rotation (to choose your angle) and a starting position.  Then for Suzanne just give her a velocity on her local X axis (or maybe Y or Z depending exactly on how you orient the parent empty) by keyframing that channel of her location.  Use x0=0 and x1 = (frame2-frame1)/fps * velocity  Pick frame1 to be the first frame you want it to be in motion, and pick frame2 such that it is after the point you want physics to to take over (which is the frame that the Animated checkbox is keyframed to become unchecked).

I used that technique to launch the cannonballs in https://youtu.be/gtP2dWmmXd8